**ABSTRACT **

PALMA PARRA, ARIADNE LIESEL. Influence of Response Spectra Definitions on the Bidirectional Seismic Behavior of Reinforced Concrete Bridge Columns. (Under the direction of Dr. Mervyn J. Kowalsky).

In recent years, researchers have debated over the appropriate definition for the

representation of earthquake demand to be used in the design of structures. Design spectra are

based on the two horizontal components of the earthquake motion recorded by sensors and are

currently generated using the RotDnn spectra definition, where “nn” refers to the percentile of the

response. This definition is independent of the in-situ sensor orientation and does not make use of

geometric means in its calculation. In this manner, the commonly used definitions are RotD50,

50th percentile or median, and RotD100, 100th percentile or maximum. The NGA-West2 research

project ground motion prediction equations (GMPEs) were based on RotD50 and are the basis for

the development of the USGS design maps. However, building codes have recently moved to using

the maximum spectra definition (RotD100) in design and concerns have been raised regarding this

decision.

Furthermore, the bridge engineering community is updating hazard maps and decisions on

spectral definition are being delayed due to a lack of understanding regarding the implications.

RotD100 response may occur in different orientations at any given period, making it very unlikely

that a single orientation of ground motion will have a response as large as the maximum spectra at

all periods, which may lead to overly conservative estimates of structural demand. Unlike previous

research that has focused on the relationship between the two response spectra definitions, this

study aims to evaluate the RotD50 and the RotD100 definitions by using the Direct

Displacement-Based Design (DDBD) approach to design SDOF RC circular columns that are then subsequently

analyzed via nonlinear time history analysis. A suite of unscaled real acceleration records were

from two sources: the Pacific Earthquake Engineering Research Center (PEER) NGA-West2

database and the Center for Engineering Strong Motion Data (CESMD).

Findings are presented as the ratio of the non-linear time history analysis displacement to

that expected from DDBD as a function of period for the two definitions of spectra. The results

confirm that the RotD100 definition is more appropriate in the design of SDOF systems while also

providing the first known verification of DDBD for bidirectional loading. On average, SDOF RC

circular columns designed to the maximum hazard definition, RotD100, show 6% less deformation

in the non-linear time history analysis than is expected from DDBD; while those designed to the

median hazard definition, RotD50, show deformations 24% greater than expected from DDBD.

Trends in the results suggest variability may be due to spectral shape effects and underestimation

of damping at short periods. Special consideration is given to the use of appropriate viscous

damping models for verification purposes. Discussion is also provided regarding how the choice

of spectra definition can impact structural design, specifically SDOF concrete columns. It is

expected that these findings will help deepen the understanding of how directionality impacts

© Copyright 2019 by Ariadne Palma

Influence of Response Spectra Definitions on the Bidirectional Seismic Behavior of Reinforced Concrete Bridge Columns

by

Ariadne Liesel Palma Parra

A thesis submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the degree of

Master of Science

Civil Engineering

Raleigh, North Carolina 2019

APPROVED BY:

_______________________________ _______________________________ Mervyn J. Kowalsky James N. Nau

Committee Chair

ii

**DEDICATION **

*Dedicated to Spencer Bailey *

iii

**BIOGRAPHY **

Ariadne L. Palma Parra was born in 1992 in Barranquilla, Colombia. She started her

undergraduate studies in 2010 at Universidad de La Salle in Bogota, Colombia, where she

completed three years of college. She transferred to North Carolina State University in 2015, where

she worked as an undergraduate research assistant and participated in the EERI Annual

Undergraduate Seismic Design Competition two years in a row. She received her BS in Civil

Engineering in the May 2017. Ariadne began her graduate research in summer 2017, along with

taking the role of SDC advisor and president of the EERI student chapter at North Carolina State

University in subsequent years. Her research interests include seismic design, analysis and

assessment of structures, and engineering seismology. After her graduate studies, she is planning

iv

**ACKNOWLEDGMENTS **

The research described in this thesis has been funded by the Alaska Department of

Transportation and Public Facilities. The support and feedback from Elmer Marx of AKDOT is

very much appreciated.

It is with immense gratitude that I acknowledge the support and guidance provided by my

advisor Dr. Mervyn J. Kowalsky. His mentoring has been invaluable throughout this process. I

also wish to thank my committee members, Dr. James Nau and Dr. Ashly Cabas Mijares for

providing their expert insight and advice.

I am thankful to my colleagues, staff, and faculty at the Constructed Facilities Laboratory

(CFL) for their continuous encouragement and making this experience enjoyable and

unforgettable. My most sincere gratitude to the graduate students that have taken the time to

discuss research and share their ideas with me.

And to my family and friends, thank you for your understanding and your faith in me,

v

**TABLE OF CONTENTS **

**LIST OF TABLES ... viii **

**LIST OF FIGURES ... ix **

**Chapter 1 : Introduction ... 1 **

1.1 Problem Description and Scope of the Research ... 1

1.2 Research Objectives ... 3

1.3 Thesis Layout ... 3

**Chapter 2 : Literature Review ... 6 **

2.1 Bidirectional Representation of Earthquake Hazard ... 6

2.1.1 Other Definitions ... 15

2.2 Earthquake Data Considerations ... 17

2.4 Damping Considerations ... 20

**Chapter 3 : Design of SDOF Bridge Columns ... 24 **

3.1 Ground Motion Data Selection ... 24

3.2 Response Spectra Definitions ... 34

3.2.1 Response Spectra and Design Spectra ... 34

3.2.2 Response Spectra Definitions: RotDnn ... 37

3.2.2 Generation of RotDnn Response Spectra ... 39

3.3 Application of DDBD ... 46

vi

3.3.2 Summary of Design Procedure ... 55

3.3.3 Design Example of a SDOF RC Circular Column ... 57

**Chapter 4 : Verification of the DDBD Method for Bidirectional Loading ... 61 **

4.1 Nonlinear Time History Analyses (NLTHA) ... 61

4.1.1 Summary of Design Verification Assumptions ... 64

4.2 NLTHA Example Calculations ... 65

4.2.1 RC Bridge Column Design - Ductility 1... 69

4.2.2 RC Bridge Column Design - Ductility 1.5... 71

4.2.3 RC Bridge Column Design - Ductility 2... 73

4.2.4 RC Bridge Column Design - Ductility 3... 75

4.2.5 RC Bridge Column Design - Ductility 4... 77

4.2.6 RC Bridge Column Design - Ductility 6... 78

4.2.7 RC Bridge Column Design - Ductility 8... 80

**Chapter 5 : Evaluation of RotD50 and RotD100 definitions ... 83 **

5.1 Displacement Ratios per Ground Motion Record ... 84

5.2 Displacement Ratios per Ductility Level ... 93

5.3 Overall Displacement Ratios and Implications for DDBD ... 102

5.4 Factors that Impact Variability in Results ... 107

5.4.1 Damping- Ductility Relationship ... 107

vii

5.4.3 Rotation Angle of Maximum Response ... 113

**Chapter 6 : Damping Model Considerations... 114 **

6.1 Damping Model for analysis of SDOF Systems ... 114

6.2.1 Initial-Stiffness versus Secant-Stiffness for a SDOF RC bridge column ... 116

6.2.2 Damping Model Comparison for Displacement Ratios Given by Median and Maximum Spectra Definitions Across Ductility Levels ... 117

**Chapter 7 : Conclusions and Recommendations ... 120 **

7.1 Summary of Outcomes ... 120

7.2 Recommendations ... 122

7.3 Future Research ... 125

**References ... 130 **

**APPENDICES ... 133 **

**APPENDIX A ... 134 **

**APPENDIX B ... 204 **

viii

**LIST OF TABLES **

Table 2.1. Geometric mean values of 𝑆𝑎𝑅𝑜𝑡𝐷100/𝑆𝑎𝑅𝑜𝑡𝐷50. Mean and standard

deviations for 𝑙𝑛𝑆𝑎𝑅𝑜𝑡𝐷100/𝑆𝑎𝑅𝑜𝑡𝐷50 estimates (Shahi & Baker, 2014). ... 13 Table 3.1: Characteristics of the suite of 65 ground motion records used in the study.

Distances provided are in kilometers, and shear wave velocities are in m/s. ... 27 Table 3.2: Displacement response for three oscillators with periods 1, 2, and 3 seconds for

the seven chosen azimuths at 30° increments. ... 41 Table 4.1: Summary of displacement ratios across seven ductility levels for 14 oscillators of

effective period of 3 seconds for RotD50 and RotD100. ... 67 Table 5.1: Characteristics of the 8 GM used for the preliminary evaluation of variability due

to spectral slope. ... 109 Table 7.1: Mean factors for each spectra definition. ... 122 Table 7.2: Approximated factors as a function of ductility level and response spectra

definition... 122 Table 7.3: Approximated (mean) factors as a function of ductility level and effective period

for each response spectra definition. ... 124 Table 7.4: Approximated (mean + 𝜎) factors as a function of ductility level and effective

period for each response spectra definition. ... 124 Table A1: Mean and variation of displacement ratios for all 65 ground motion record pairs

for the median response spectra definition, RotD50, shown at seven ductility

levels and per record. ... 135 Table A2: Mean and variation of displacement ratios for 65 ground motion record pairs for

ix

**LIST OF FIGURES **

Figure 2.1: GMRotI50 and GMRotDnn at 0th, 50th, and 100th percentiles (Boore et al.,

2006). ... 8 Figure 2.2: Example of acceleration orbit of a 2-degree-of-freedom oscillator used to

compute minimum and maximum spectral demand (Huang et al, 2008). ... 9 Figure 2.3: Displacement response spectra at 5% damping ratio given by RotDnn and RotInn

definitions at 50th and 100th percentiles for the 1978 Tabas earthquake (Boore, 2010). ... 11 Figure 2.4: Plan views of typologies considered by Nievas and Sullivan (2017). ... 12 Figure 2.5: Comparison of models for geometric mean of the ratio SaRotD100/SaRotD50

(Shahi & Baker, 2014). ... 14 Figure 2.6: Comparison of the ratio of 𝑆𝑎𝑅𝑜𝑡𝐷100 to 𝑆𝑎𝑅𝑜𝑡𝐷50 observed for the

Canterbury ground motion data with the model of Shahi and Baker based on

worldwide data (Bradley & Baker, 2014). ... 15 Figure 2.7: Magnitude-distance distribution of strong-motion records in the NGA West2

database (Ancheta et al., 2013). ... 17 Figure 2.8: Elastic Response Spectra reduced for equivalent damping values (Priestley et al.,

2007). ... 21 Figure 2.9: RC column experimental test schematic and shake table set up (Petrini et al.,

2008). ... 22 Figure 2.10: Displacement response of an experimental RC column test compared to

responses given by models with different stiffness assumptions (Petrini et al,

2007). ... 23 Figure 3.1: Sources of motion data for the chosen 65 record pairs. ... 24 Figure 3.2: Distribution of some ground motion parameters for the suite of selected 65

ground motion records. ... 25 Figure 3.3: Location of stations of the 4 records selected from the 2019 Ridgecrest, CA

earthquake (Mw7.1). Color scheme indicates intensity of the motion in terms of percentage of gravity (CESMD). ... 29 Figure 3.4: Location of stations of the 5 records selected from the 2018 Anchorage, AK

earthquake (Mw7.1). Color scheme indicates intensity of the motion in terms of percentage of gravity (CESMD). ... 30 Figure 3.5: Distribution of distances for the magnitude range of the suite of 65 ground motion

records. ... 31 Figure 3.6: Distribution of distances for the suite of 65 ground motion records. ... 31 Figure 3.7: Distribution of PGA values for the magnitude range of the 65 ground motion

x Figure 3.9: Generation of response spectrum (Part b adapted from Hachem, 2004). ... 35 Figure 3.10: Design Spectra for RotD50 and RotD100 definitions. ... 36 Figure 3.11: Displacement Response Spectra for each individual as-recorded component,

SD1 and SD2, and for the median and maximum definition, RotD50 and

RotD100. ... 37 Figure 3.12: Rotation angles as a function of effective period corresponding to the RotD100

response spectrum with 5% damping ratio. ... 38 Figure 3.13: Combination of a pair of orthogonal as-recorded acceleration time series for a

particular rotation angle and the corresponding response spectrum for the

resultant time series. ... 40 Figure 3.14: Range of azimuths from 0° to 180° with an angle increment of 30°. ... 40 Figure 3.15: RotDnn displacement response spectra at 00th, 50th, and 100th percentiles for

oscillators with fundamental periods of 1s, 2s, and 3s under the 2011

Christchurch earthquake record at the Kaipoi North School station. ... 42 Figure 3.16: Displacement response spectra for the two original orthogonal components, and

for the two Rotnn definitions, RotD50 and RotD100. ... 43 Figure 3.17: Displacement Response Spectra for the suite 65 pairs of ground motion records. . 44 Figure 3.18: Initial and Secant Stiffness characterization of hysteretic response (Priestley et

al., 2007) ... 46 Figure 3.19: Flowchart comparing Force-based Design and Direct Displacement-based

Design methodologies (Priestley et al., 2007). ... 47 Figure 3.20: Displacement Response Spectra Generation. ... 48 Figure 3.21: Application of DDBD method. ... 49 Figure 3.22: Damping - Ductility relationships for different hysteresis rules (Priestley et al.,

2017). ... 50 Figure 3.23: Damping - Ductility relationship for selected ductility levels using Takeda Thin

(TT) hysteresis. ... 51 Figure 3.24: Strength of an idealized SDOF bridge column under design earthquake at the

chosen target displacement. ... 53 Figure 3.25: Summary of complete application of DDBD method for the design of RC bridge

columns... 54 Figure 3.26: SDOF concrete column design procedure using DDBD for a particular RotDnn

Definition. ... 56 Figure 3.27: Design Example Schematic ... 57 Figure 3.28: Displacement Response Spectra at ductility level of 2( ξ = 12.7%) for 1995

xi Figure 4.1: Schematic of idealized RC column design under corresponding pair of

as-recorded motions, where Δmax is the maximum radial displacement from the origin... 62 Figure 4.2: Modified Takeda Hysteresis rule with Emori Unloading (Carr, 2017). ... 63 Figure 4.3: As-recorded pair of acceleration histories for 1992 Landers, CA earthquake at

Lucerne Station ... 65 Figure 4.4: Target Displacements at effective periods equal to 3 seconds for RotD50 and

RotD100 across 7 ductility levels. ... 66 Figure 4.5: NLTHA Results Example of RC column designed for ductility level 1 using

RotD100 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 69 Figure 4.6: NLTHA Results Example of RC column designed for ductility level 1 using

RotD50 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 70 Figure 4.7: NLTHA Results Example of RC column designed for ductility level 1.5 using

RotD100 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 72 Figure 4.8: NLTHA Results Example of RC column designed for ductility level 1.5 using

RotD50 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 73 Figure 4.9: NLTHA Results Example of RC column designed for ductility level 2 using

RotD100 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 74 Figure 4.10: NLTHA Results Example of RC column designed for ductility level 2 using

RotD50 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 75 Figure 4.11: NLTHA Results Example of RC column designed for ductility level 3 using

RotD100 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 76 Figure 4.12: NLTHA Results Example of RC column designed for ductility level 3 using

RotD50 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 76 Figure 4.13: NLTHA Results Example of RC column designed for ductility level 4 using

RotD100 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 77 Figure 4.14: NLTHA Results Example of RC column designed for ductility level 4 using

RotD50 spectrum of 1992 Landers earthquake record at Lucerne station

xii Figure 4.15: NLTHA Results Example of RC column designed for ductility level 6 using

RotD100 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 79 Figure 4.16: NLTHA Results Example of RC column designed for ductility level 6 using

RotD50 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 80 Figure 4.17: NLTHA Results Example of RC column designed for ductility level 8 using

RotD100 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 81 Figure 4.18: NLTHA Results Example of RC column designed for ductility level 8 using

RotD50 spectrum of 1992 Landers earthquake record at Lucerne station

(RSN879). ... 81 Figure 5.1: RotD50 and RotD100 Elastic Response Spectra (ξ = 5%) for 3 pairs of ground

motion records from the NGA West2 database: the 1992 Landers earthquake (RSN879), the 1999 Chi Chi earthquake (RSN1231), and the 2011 Christchurch earthquake (RSN8099). ... 84 Figure 5.2: Sample of acceleration histories for 3 pairs of records used in the study. ... 85 Figure 5.3: Displacement ratios across all seven ductility level observed for the 1992

Landers, CA earthquake record (RSN879). ... 87 Figure 5.4: Displacement ratios across all seven ductility level observed for the 1999 Chi

Chi, Taiwan earthquake record (RSN1231). ... 88 Figure 5.5: Displacement ratios across all seven ductility level observed for the 2011

Christchurch, NZ earthquake record (RSN8099). ... 89 Figure 5.6: Mean and coefficient of variation of the displacement ratios found for the two

response spectra definitions, RotD50 and RotD100, per each of the 65

earthquake records. ... 91 Figure 5.7: Response spectra and displacement ratio results for the 1935 Helena, MT record

(Mw6). ... 92 Figure 5.8: Displacement ratios and coefficient of variation per ductility level as a function of

period for the median (RotD50) and maximum (RotD100) response spectra

definitions. ... 94 Figure 5.9: Geometric mean and coefficient of variation of the displacement ratios as a

function of ductility level for the two response spectra definitions, RotD50 and RotD100. ... 99 Figure 5.10: Empirical data distribution per ductility level for displacement ratios given by

each response spectra definition, RotD50 and RotD100. Each curve corresponds to displacement ratios of 3,250 RC columns. ... 100 Figure 5.11: Normal distribution per ductility level for displacement ratios given by each

xiii Figure 5.12: Displacement ratios averaged across all ductility levels a function of period for

the two response spectra definitions, RotD50 and RotD100. Mean and coefficient of variation across periods is given for 22,750 RC columns designed to each

definition... 102 Figure 5.13: Cumulative distribution function based on the empirical displacement ratios for

each definition, RotD50 and RotD100, averaged across all ductility levels (1 to 8). Each curve corresponds to displacement ratios of 22,750 RC columns. ... 103 Figure 5.14: Cumulative distribution function based on the empirical displacement ratios of

each definition, RotD50 and RotD100, averaged across all ductility levels >1 (1.5 to 8). Each curve corresponds to displacement ratios of 19,500 RC columns.104 Figure 5.15: Ratios of maximum inelastic response (Δmax) to that predicted by DDBD (Δt)

for ductility levels 1 and 8 for designs under the 1992 Landers record at the

Lucerne station (RSN879). ... 105 Figure 5.16: Period dependency of hysteretic component of equivalent viscous damping

(Priestley et al., 2007). ... 107 Figure 5.17: Spectral Shapes for a group of 8 ground motions of Mw > 6.5. ... 111 Figure 5.18: Evaluation of spectral shape influence for 8 GM. The normalized spectra for the

group of ground motions is shown with the corresponding amount of RC column that had ratios below 0.6. ... 112 Figure 6.1: Inelastic response of an SDOF oscillator with “Thin” modified Takeda hysteresis

(Priestley et al., 2007). ... 115 Figure 6.2: Comparison of damping force hysteresis for two oscillators with an effective

period of 3 seconds and designed to μ = 2 using different damping model

definitions. ... 116 Figure 6.3: Displacement response history for two oscillators with different damping models.117 Figure 6.4: Damping model comparison for displacement ratios given by median and

maximum spectra definitions across ductility levels. ... 118 Figure 7.1: Approximated factors as a function of ductility level and response spectra

definition... 123 Figure 7.2: Approximated (mean) factors as a function of ductility and effective period at a

0.25s interval. ... 123 Figure 7.3: NLTHA analysis results for a RC column of effective period equal to 3 seconds

at μ = 1, where the rotation angle at the direction of maximum response matches the predicted rotation angle (160°)at 3s given by the RotD100 spectrum of the corresponding ground motion record (RSN879). ... 126 Figure 7.4: Hysteresis Rules for Nonlinear Time History Analysis. ... 126 Figure 7.5: Design spectra given by the current AASHTO code and the new proposed 2018

xiv Figure 1: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the NSMP8027 record for 2018 Anchorage, AK earthquake. ... 139 Figure 2: Displacement ratios across all seven ductility level observed for the 2018

Anchorage, AK earthquake record (NSMP8027). ... 139 Figure 3: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the NSMP8036 record for 2018 Anchorage, AK earthquake. ... 140 Figure 4: Displacement ratios across all seven ductility level observed for the 2018

Anchorage, AK earthquake record (NSMP8036). ... 140 Figure 5: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the NSMP8037 record for 2018 Anchorage, AK earthquake. ... 141 Figure 6: Displacement ratios across all seven ductility level observed for the 2018

Anchorage, AK earthquake record (NSMP8037). ... 141 Figure 7: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the NSMP8038 record for 2018 Anchorage, AK earthquake. ... 142 Figure 8: Displacement ratios across all seven ductility level observed for the 2018

Anchorage, AK earthquake record (NSMP8038). ... 142 Figure 9: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the NSMP8047 record for 2018 Anchorage, AK earthquake. ... 143 Figure 10: Displacement ratios across all seven ductility level observed for the 2018

Anchorage, AK earthquake record (NSMP8047). ... 143 Figure 11: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN2 record for 1935 Helena, MT earthquake. ... 144 Figure 12: Displacement ratios across all seven ductility level observed for the 1935 Helena,

MT earthquake record (RSN2). ... 144 Figure 13: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN6 record for 1940 Imperial Valley, CA earthquake. ... 145 Figure 14: Displacement ratios across all seven ductility level observed for the 1940

Imperial Valley, CA earthquake record (RSN6). ... 145 Figure 15: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN8 record for the 1941 Northern California earthquake. ... 146 Figure 16: Displacement ratios across all seven ductility level observed for the 1952

Northern California earthquake record (RSN8). ... 146 Figure 17: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN15 record for the 1952 Kern County, CA earthquake. ... 147 Figure 18: Displacement ratios across all seven ductility level observed for the 1952 Kern

County, CA earthquake record (RSN15)... 147 Figure 19: Displacement Response Spectra and distribution of rotation angles for RotD100

xv Figure 20: Displacement ratios across all seven ductility level observed for the 1954

Northern California earthquake record (RSN20). ... 148 Figure 21: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN22 record for the 1956 El Alamo earthquake. ... 149 Figure 22: Displacement ratios across all seven ductility level observed for the 1956 El

Alamo earthquake record (RSN22). ... 149 Figure 23: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN57 record for the 1971 San Fernando, CA earthquake. ... 150 Figure 24: Displacement ratios across all seven ductility level observed for the 1971 San

Fernando, CA earthquake record (RSN57). ... 150 Figure 25: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN95 record for 1972 Managua, Nicaragua earthquake. ... 151 Figure 26: Displacement ratios across all seven ductility level observed for the 1972

Managua, Nicaragua earthquake record (RSN95). ... 151 Figure 27: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN125 record for 1976 Friuli, Italy earthquake. ... 152 Figure 28: Displacement ratios across all seven ductility level observed for the 1976 Friuli,

Italy earthquake record (RSN125). ... 152 Figure 29: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN126 record for 1976 Gazli, USSR earthquake. ... 153 Figure 30: Displacement ratios across all seven ductility level observed for the 1976 Gazli,

USSR earthquake record (RSN126). ... 153 Figure 31: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN143 record for 1978Tabas, Iran earthquake. ... 154 Figure 32: Displacement ratios across all seven ductility level observed for the 1978 Tabas,

Iran earthquake record (RSN143). ... 154 Figure 33: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN149 record for 1979 Coyote Lake, CA earthquake. ... 155 Figure 34: Displacement ratios across all seven ductility level observed for the 1979 Coyote

Lake, CA earthquake record (RSN149). ... 155 Figure 35: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the 1979 Imperial Valley, CA earthquake. ... 156 Figure 36: Displacement ratios across all seven ductility level observed for the 1979

Imperial Valley, CA earthquake record (RSN170). ... 156 Figure 37: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN174 record for 1979 Imperial Valley, CA earthquake. ... 157 Figure 38: Displacement ratios across all seven ductility level observed for the 1979

xvi Figure 39: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN230 record for 1980 Mammoth Lakes, CA earthquake. ... 158 Figure 40: Displacement ratios across all seven ductility level observed for the 1980

Mammoth Lakes, CA earthquake record (RSN230). ... 158 Figure 41: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN265 record for 1980 Victoria, Mexico earthquake. ... 159 Figure 42: Displacement ratios across all seven ductility level observed for the 1980

Victoria, Mexico earthquake record (RSN265). ... 159 Figure 43: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN292 record for 1980 Irpinia, Italy earthquake. ... 160 Figure 44: Displacement ratios across all seven ductility level observed for the 1980 Irpinia,

Italy earthquake record (RSN292). ... 160 Figure 45: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN319 record for 1981 Westmorland, CA earthquake. ... 161 Figure 46: Displacement ratios across all seven ductility level observed for the 1981

Westmorland, CA earthquake record (RSN319). ... 161 Figure 47: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN367 record for 1983 Coalinga, CA earthquake. ... 162 Figure 48: Displacement ratios across all seven ductility level observed for the 1983

Coalinga, CA earthquake record (RSN367). ... 162 Figure 49: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN368 record for 1983 Coalinga, CA earthquake. ... 163 Figure 50: Displacement ratios across all seven ductility level observed for the 1983

Coalinga, CA earthquake record (RSN368). ... 163 Figure 51: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN405 record for 1983 Coalinga, CA earthquake. ... 164 Figure 52: Displacement ratios across all seven ductility level observed for the 1983

Coalinga, CA earthquake record (RSN405). ... 164 Figure 53: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN411 record for 1983 Coalinga, CA earthquake. ... 165 Figure 54: Displacement ratios across all seven ductility level observed for the 1983

Coalinga, CA earthquake record (RSN411). ... 165 Figure 55: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN595 record for 1983 Coalinga, CA earthquake. ... 166 Figure 56: Displacement ratios across all seven ductility level observed for the 1987

Whittier Narrows, CA earthquake record (RSN595). ... 166 Figure 57: Displacement Response Spectra and distribution of rotation angles for RotD100

xvii Figure 58: Displacement ratios across all seven ductility level observed for the 1987

Whittier Narrows, CA earthquake record (RSN615). ... 167 Figure 59: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN642 record for 1987 Whittier Narrows, CA earthquake. ... 168 Figure 60: Displacement ratios across all seven ductility level observed for the 1987

Whittier Narrows, CA earthquake record (RSN642). ... 168 Figure 61: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN652 record for 1987 Whittier Narrows, CA earthquake. ... 169 Figure 62: Displacement ratios across all seven ductility level observed for the 1987

Whittier Narrows, CA earthquake record (RSN652). ... 169 Figure 63: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN668 record for 1987 Whittier Narrows, CA earthquake. ... 170 Figure 64: Displacement ratios across all seven ductility level observed for the 1987

Whittier Narrows, CA earthquake record (RSN668). ... 170 Figure 65: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN728 record for 1987 Superstition Hills, CA earthquake. ... 171 Figure 66: Displacement ratios across all seven ductility level observed for the 1987

Superstition Hills, CA earthquake record (RSN728). ... 171 Figure 67: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN738 record for 1989 Loma Prieta, CA earthquake. ... 172 Figure 68: Displacement ratios across all seven ductility level observed for the 1989 Loma

Prieta, CA earthquake record (RSN738). ... 172 Figure 69: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN758 record for 1989 Loma Prieta, CA earthquake. ... 173 Figure 70: Displacement ratios across all seven ductility level observed for the 1989 Loma

Prieta, CA earthquake record (RSN758). ... 173 Figure 71: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN777 record for 1989 Loma Prieta, CA earthquake. ... 174 Figure 72: Displacement ratios across all seven ductility level observed for the 1989 Loma

Prieta, CA earthquake record (RSN777). ... 174 Figure 73: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN779 record for 1989 Loma Prieta, CA earthquake. ... 175 Figure 74: Displacement ratios across all seven ductility level observed for the 1989 Loma

Prieta, CA earthquake record (RSN779). ... 175 Figure 75: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN799 record for 1989 Loma Prieta, CA earthquake. ... 176 Figure 76: Displacement ratios across all seven ductility level observed for the 1989 Loma

xviii Figure 77: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN879 record for 1992 Landers, CA earthquake. ... 177 Figure 78: Displacement ratios across all seven ductility level observed for the 1992 Landers,

CA earthquake record (RSN879). ... 177 Figure 79: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1004 record for 1994 Northridge, CA earthquake. ... 178 Figure 80: Displacement ratios across all seven ductility level observed for the 1994

Northridge, CA earthquake record (RSN1004). ... 178 Figure 81: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1084 record for the 1994 Northridge, CA earthquake. ... 179 Figure 82: Displacement ratios across all seven ductility level observed for the RSN1084

record for the 1994 Northridge, CA earthquake. ... 179 Figure 83: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1110 record for 1995 Kobe, Japan earthquake. ... 180 Figure 84: Displacement ratios across all seven ductility level observed for the 1995 Kobe,

Japan earthquake record (RSN1110). ... 180 Figure 85: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1120 record for 1995 Kobe, Japan earthquake. ... 181 Figure 86: Displacement ratios across all seven ductility level observed for the 1995 Kobe,

Japan earthquake record (RSN1120). ... 181 Figure 87: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1158 record for 1999 Kocaeli, Turkey earthquake. ... 182 Figure 88: Displacement ratios across all seven ductility level observed for the 1999

Kocaeli, Turkey earthquake record (RSN1158). ... 182 Figure 89: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1231 record for 1995 Chi Chi, Taiwan earthquake. ... 183 Figure 90: Displacement ratios across all seven ductility level observed for the 1999 Chi Chi,

Taiwan earthquake record (RSN1231). ... 183 Figure 91: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1626 record for 1972 Sitka, AK earthquake. ... 184 Figure 92: Displacement ratios across all seven ductility level observed for the 1972 Sitka,

AK earthquake record (RSN1626). ... 184 Figure 93: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN1628 record for 1979 Stelias, AK earthquake. ... 185 Figure 94: Displacement ratios across all seven ductility level observed for the 1979 Stelias,

AK earthquake record (RSN1628). ... 185 Figure 95: Displacement Response Spectra and distribution of rotation angles for RotD100

xix Figure 96: Displacement ratios across all seven ductility level observed for the 1979 Stelias,

AK earthquake record (RSN1629). ... 186 Figure 97: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN2107 record for 2002 Denali, AK earthquake. ... 187 Figure 98: Displacement ratios across all seven ductility level observed for the 2002 Denali,

AK earthquake record (RSN2107). ... 187 Figure 99: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN2109 record for 2002 Denali, AK earthquake. ... 188 Figure 100:Displacement ratios across all seven ductility level observed for the 2002 Denali,

AK earthquake record (RSN2109). ... 188 Figure 101: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN2110 record for 2002 Denali, AK earthquake. ... 189 Figure 102:Displacement ratios across all seven ductility level observed for the 2002 Denali,

AK earthquake record (RSN2110). ... 189 Figure 103: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN2113 record for 2002 Denali, AK earthquake. ... 190 Figure 104:Displacement ratios across all seven ductility level observed for the 2002 Denali,

AK earthquake record (RSN2113). ... 190 Figure 105: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN2114 record for 2002 Denali, AK earthquake. ... 191 Figure 106:Displacement ratios across all seven ductility level observed for the 2002 Denali,

AK earthquake record (RSN2114). ... 191 Figure 107: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN2115 record for 2002 Denali, AK earthquake. ... 192 Figure 108:Displacement ratios across all seven ductility level observed for the 2002 Denali,

AK earthquake record (RSN2115). ... 192 Figure 109: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN3835 record for the 2002 Denali, AK earthquake. ... 193 Figure 110:Displacement ratios across all seven ductility level observed for the 2002 Denali,

AK earthquake record (RSN3835). ... 193 Figure 111: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN4070 record for the 2004 Parkfield, CA earthquake. ... 194 Figure 112:Displacement ratios across all seven ductility level observed for the 2004

Parkfield, CA earthquake record (RSN4070). ... 194 Figure 113: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN4107 record for the 2004 Parkfield, CA earthquake. ... 195 Figure 114:Displacement ratios across all seven ductility level observed for the 2004

xx Figure 115: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN4116 record for the 2004 Parkfield, CA earthquake. ... 196 Figure 116:Displacement ratios across all seven ductility level observed for the 2004

Parkfield, CA earthquake record (RSN4116). ... 196 Figure 117: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN4134 record for the 2004 Parkfield, CA earthquake. ... 197 Figure 118:Displacement ratios across all seven ductility level observed for the 2004

Parkfield, CA earthquake record (RSN4134). ... 197 Figure 119: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN5672 record for the 2008 Iwate, Japan earthquake. ... 198 Figure 120:Displacement ratios across all seven ductility level observed for the 2008 Iwate,

Japan earthquake record (RSN5672). ... 198 Figure 121: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the RSN8099 ... 199 Figure 122: Displacement ratios across all seven ductility level observed for the 2011

Christchurch, NZ earthquake record (RSN8099). ... 199 Figure 123: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the SCSN-CCC record for 2019 Ridgecrest, CA earthquake. ... 200 Figure 124:Displacement ratios across all seven ductility level observed for the 2019

Ridgecrest, CA earthquake record (SCSN-CCC). ... 200 Figure 125: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the SCSN-CLC record for 2019 Ridgecrest, CA earthquake. ... 201 Figure 126:Displacement ratios across all seven ductility level observed for the 2019

Ridgecrest, CA earthquake record (SCSN-CLC). ... 201 Figure 127: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the SCSN-SRT record for 2019 Ridgecrest, CA earthquake. ... 202 Figure 128:Displacement ratios across all seven ductility level observed for the 2019

Ridgecrest, CA earthquake record (SCSN-SRT). ... 202 Figure 129: Displacement Response Spectra and distribution of rotation angles for RotD100

spectrum for the SCSN-TOW2 record for 2019 Ridgecrest, CA earthquake. ... 203 Figure 130:Displacement ratios across all seven ductility level observed for the 2019

Ridgecrest, CA earthquake record (TOW2). ... 203 Figure 131: Displacement ratios and coefficient of variation per ductility level as a function

of period for the median (RotD50) and maximum (RotD100) response spectra definitions. ... 205 Figure 132: Geometric mean and coefficient of variation of the displacement ratios as a

xxi Figure 133:Displacement ratios averaged across all ductility levels a function of period for

the two response spectra definitions, RotD50 and RotD100. Mean and coefficient of variation across periods is given for 22,750 RC columns designed to each

definition... 209 Figure 134: Cumulative distribution function based on the empirical displacement ratios for

each definition, RotD50 and RotD100, averaged across all ductility levels (1 to 8). Each curve corresponds to displacement ratios of 22,750 RC columns. ... 210 Figure 135: Cumulative distribution function based on the empirical displacement ratios of

each definition, RotD50 and RotD100, averaged across all ductility levels >1 (1.5 to 8). Each curve corresponds to displacement ratios of 19,500 RC columns.210 Figure 136: Empirical data distribution per ductility level for displacement ratios given by

each response spectra definition, RotD50 and RotD100. Each curve corresponds to displacement ratios of 3,250 RC columns. ... 211 Figure 137: Normal distribution per ductility level for displacement ratios given by each

response spectra definition, RotD50 and RotD100. Each curve corresponds to displacement ratios of 3,250 RC columns. ... 211

1

**Chapter 1**

**: Introduction **

**1.1 Problem Description and Scope of the Research **

This study investigates the response spectra definitions of the earthquake hazard that have

been incorporated in design codes and are currently used in practice. Ground motions are typically

recorded by triaxial accelerometers that measure acceleration in three orthogonal directions: two

in the horizontal plane and one in the vertical. The azimuths of the horizontal sensors are arbitrary.

Design spectra for a specific location are based on historical bi-directional horizontal acceleration

records. For a specific ground motion recording, the response spectrum can be generated for each

component. However, it is necessary to account for all possible angles in which the ground motion

could occur because the intensity of ground shaking is not uniform across different orientations

(Boore et al., 2006). Several methods have been proposed to combine the horizontal acceleration

components to find a representative response spectrum of the ground motion. These include

response spectra methods that use the geometric mean of the two horizontal components rotated

in small angle increments over a 90° range, such as GMRotDnn and GMRotInn (Boore et al.,

2006). The currently used definition is known as RotDnn. This response spectrum definition is

independent of the in situ orientation of the sensors by taking into account the full range of spectral

amplitudes over all possible rotation angles, and it can represent any percentile without use of

geometric means (Boore, 2010). The two percentiles used herein are the median, RotD50, and the

maximum, RotD100. Recent changes in building codes motivate the need for a numerical analysis

of the impact of using a specific definition in bridge design.

The NGA-West2 research program has developed models that predict the median spectral

acceleration of ground motions, also referred to as RotD50 acceleration response spectra (Ancheta

2 RotD100, as the definition of the hazard for design (Stewart et al., 2011). Research has been done

to determine factors to convert RotD50 spectra to RotD100 spectra for incorporation in design

codes, such as the ASCE 7 standard and the NEHRP provisions (Shahi & Baker, 2014; BSSC,

2009). However, concerns have been raised regarding the choice of the maximum definition, given

that the principal axes of typical buildings would not both align with the maximum-direction

during an earthquake (Stewart et al., 2011). Only SDOF structural systems would not have

preferred directions of response while having the same dynamic properties in all possible

horizontal directions, while multiple-degree-of-freedom (MDOF) systems can have varying

dynamic properties depending on their principal axes. Due to this, further investigation is

necessary to be able to reach a consensus in the engineering community.

This problem has been studied with a focus on hazard definition but little research has been

done on the implications for structural design. This study aims to fill this gap, and is motivated by

the need for adequately investigating the nonlinear response of bridge structures to bidirectional

loading to reach a consensus on what definition of response spectra to use in design. Damage can

be related to levels of strain or drifts, and so it can be directly related to displacement. Because of

this, the designs in this study were done following the Direct Displacement-Based Design (DDBD)

approach (Priestley et al., 2007). In summary, this method starts with selection of a target

displacement for the desired limit state and ends with the required strength of the structure. It has

been shown that the DDBD approach compares well with Nonlinear Time History Analyses

(NLTHA) for seismic loading in one direction (Priestley et al., 2007). However, verification

3

**1.2 Research Objectives **

To address the gap recognized above, this research aims to evaluate the response of SDOF

bridge structures, specifically reinforced concrete circular columns, when they are designed for

seismic hazards represented by response spectra given by RotD50 and RotD100. Furthermore, this

research provides verification of the DDBD approach for bidirectional loading. In addition, this

research is expected to heighten the awareness within the structural engineering community of the

potential structural performance implications that come with the choice of response spectra

definition.

**1.3 Thesis Layout **

Chapter 1 contains a general introduction to the research. It describes the scope and

motivation of this study. The research objectives and organization of this thesis are also included

in this chapter.

Chapter 2 presents a review of the related work on the subject of the definition of response

spectrum and how it has evolved throughout the years. It also provides a discussion on the selection

criteria, processing and characterization of strong-motion records for use in nonlinear time history

analyses. The chapter ends with a section describing the necessary damping model considerations

for inelastic analysis of SDOF structures.

Chapter 3 describes how the ground motion data were selected for these analyses. This is

followed by a discussion on response spectra definitions, specifically focusing on how to generate

the RotDnn response spectra for the maximum and median percentiles, RotD100 and RotD50

4 definitions and the application of the Direct Displacement-Based Design methodology for the

design of circular reinforced concrete (RC) bridge columns.

Chapter 4 outlines the steps and assumptions needed to perform nonlinear time history

analysis (NLTHA) in this study for the verification of the DDBD approach for bidirectional

earthquake loading of SDOF RC circular bridge columns. It includes a discussion on the results of

the designs and analysis of 14 SDOF columns as a means to illustrate typical outcomes achieved

for all SDOF systems considered in this study.

Chapter 5 discusses the observed results in terms of displacement ratios (Δ_{𝑚𝑎𝑥}/Δ_{𝑡}) for each

of the as-recorded ground motion pairs to evaluate how well each response spectra definition

(RotD50 and RotD100) defines the structural response. Overall findings are then presented for the

complete suite of records (65 GM) for seven ductility levels. Average values and coefficient of

variation are provided as a function of period. This is followed by an assessment of the

displacement response across all ductility levels for each response spectra definition and its

implications in seismic design, specifically for the DDBD approach. The chapter concludes with

a discussion on the factors that affect the variability of the results.

Chapter 6 describes the need for adequately defining the damping model for inelastic

analysis of structures. It focuses on how to characterize the terms in the Rayleigh Damping Model,

specifically the stiffness proportional term. It presents a comparison of the structural response for

2 SDOF structures modelled with different stiffness assumptions, followed by a discussion on how

this decision can impact the results observed when evaluating the response spectra definitions as

5 Chapter 7 presents a summary of the main outcomes and conclusions. Recommendations

are provided for the use of response spectra definition in the design of SDOF RC circular columns.

The chapter ends with a brief discussion of the specific topics where future work is still needed.

Finally, the previous chapters are complemented with three appendices. Appendix A

includes response spectra and displacement ratios for the two spectra definitions (RotD50 and

RotD100) for each record utilized in this study. Appendix B includes overall displacement ratio

results as well as per ductility level when using initial stiffness in the Rayleigh Damping Model

for the nonlinear time history analyses. Appendix C includes the MATLAB code used for the

generation of the RotDnn response spectra and subsequent design of 700 SDOF systems with the

application of the DDBD method for each pair record. In addition, example input and output

6

**Chapter 2**

**: Literature Review **

The objective of this chapter is to outline the evolution of thought in the area of seismic

design, from the perspectives of earth scientists and structural engineers, with respect to (1) how

response spectra has been defined over the years, (2) how strong-motion records are characterized

and processed, as well as (3) how SDOF structures are analyzed and modeled for bidirectional

loading, and the (4) necessary damping model considerations. The information presented herein

aims to provide the necessary context for one to be able to understand the background and need of

the research performed in this study and educate on the current state of practice.

**2.1 Bidirectional Representation of Earthquake Hazard **

Earthquake engineering is a branch of engineering that is highly multidisciplinary. As a

consequence, interdisciplinary communication is essential. In 2006, Baker and Cornell were

among the first to recognize that miscommunication between earth scientists and structural

engineers can lead to systematic errors that may result in non-conservative solutions. They

acknowledged the fact that the two communities defined the pseudo spectral acceleration

differently. Pseudo spectral acceleration (𝑆𝑎), shown in Equation 2.1, is equal to the spectral

displacement times the square of the natural frequency and it is the most commonly used intensity

measure for representation of the earthquake hazard when depicted via a response spectrum.

𝑺_{𝒂} = 𝑺_{𝒅}𝒘𝟐

*Equation 2.1 *

As such, Baker and Cornell (2006) highlight the need to use a consistent definition of this

parameter in the assessment of structures under seismic loading. They indicate that seismologists

and geotechnical engineers perform ground motion hazard analyses using the geometric mean of

spectral acceleration of two components (𝑆𝑎𝑔.𝑚.), as used in the attenuation models at the time;

7
arbitrary component (𝑆𝑎_{𝑎𝑟𝑏}), equivalent to using a single component of a ground motion. This

disparity could influence results greatly. For instance, in a 2D analysis, using 𝑆𝑎_{𝑔.𝑚.} would result

in lower target spectra than when using 𝑆𝑎𝑎𝑟𝑏 but with higher dispersion in the structural response.

As such, not reconciling these two measures in analysis would introduce unnecessary variation

into the relationship between hazard and response. Baker and Cornell (2006) suggest three

alternative methods to current practice at the time that essentially maintain the same IM for both

hazard and response, 𝑆𝑎_{𝑔.𝑚.} or 𝑆𝑎_{𝑎𝑟𝑏}, or correct the results to reflect the use of different

definitions. Furthermore, they suggested the creation of attenuation models for both IMs as it

would provide necessary flexibility in the definition of spectral acceleration to be used in analysis.

In the following years, advances in the area resulted in different representations of the

hazard. Boore et al (2006) found that the geometric mean of the as-recorded motions was sensitive

to the orientation of the sensors installed in the field, and as such the measure of ground motion

intensity could differ for the same actual ground motion. For example, if the ground motion was

linearly polarized, and one of the sensors happened to be aligned in the orthogonal direction, the

response spectrum for the recording of this sensor would be zero, which would make the geometric

mean equal to zero, irrespective of the amplitude. With this in mind, the authors introduced two

new measures that aimed to replace the use of the geometric mean of the response spectra as the

response variable in the prediction of strong ground motions. These two new measures were

GMRotDpp and GMRotIpp, both being orientation-independent with regards to the two horizontal

8

Figure 2.1: GMRotI50 and GMRotDnn at 0th, 50th, and 100th percentiles (Boore et al., 2006).

GMRotDpp is defined as a specific percentile value (pp) of the set of geometric means

obtained for all nonredundant rotation angles between 0 and 90 degrees for a specific oscillator

period. Boore et al. (2006) deemed the GMRotDpp dependence on period as an unappealing

characteristic given that it could “obscure the physical interpretation of the measure.” On the other

hand, GMRotIpp is similar to GMRotDpp in its calculation, but it corresponds to the geometric

mean response spectra of the two components after a single rotation angle for all oscillators’

periods, which minimizes the spread of rotation-dependent geometric mean over the period range.

The two measures are compared in Figure 2.1 for a particular pair of ground motion recording

during the 1971 San Fernando earthquake. As it can be observed, the two measures, GMRotD50

and GMRotI50, are very similar to each other. The 50th percentile of the latter, or GMRotI50,

was adopted as the ground motion intensity measure used in the ground motion prediction

equations (GMPEs) developed during the Pacific Earthquake Engineering Research (PEER)

center’s Next Generation Attenuation of Ground Motions (NGA) Project, which concluded in

9 sets of ground motion attenuation models created. All five models used the GMRotI50 spectra

definition, a median measure of ground motion intensity.

Figure 2.2: Example of acceleration orbit of a 2-degree-of-freedom oscillator used to compute minimum and maximum spectral demand (Huang et al, 2008).

However, some engineers expressed interest in a definition that represented a maximum

measure. In 2008, Huang et al. argued that maximum demands could substantially surpass the

GMRotI50 demands used in the NGA West Project, specifically in the near-fault region, which

could impact the seismic maps and consequently have implications for seismic design and

assessment of structures. The authors calculated the maximum and minimum spectral demands for

a structure of a certain period subjected to a pair of ground motion records. As shown in Figure

2.2, the acceleration orbit of the oscillator can help identify the maximum acceleration as the

farthest point from the origin and its corresponding orientation; as well as the minimum

acceleration and its orientation. They identified the maximum acceleration for a range of periods

for 147 pairs of records to find the ratio of the maximum spectral demand to the demand given by

GMRotI50. In sum, the proposed scale factors were 1.1 and 1.3 for the spectral response

accelerations at 0.2 seconds and 1.0 second, respectively. Based on this research, the Building

10 ASCE 7-10 to convert the median measure representation of the ground motion to a

“maximum-direction” definition.

In response to this concern, Boore (2010) introduced new measures of response spectra

definitions that had two goals: (1) measures that could represent any percentile of the response

without the use of geometric means and (2) still be independent of in situ orientation. This meant

that a maximum representation of the spectral demand, as well as the median was possible. These

new measures are called RotInn and RotDnn, where “nn” refers to the percentile of the response.

Similar to previous measures, the former is a period-independent-rotation angle measure with the

goal of avoiding variation over a broad range of periods; and the latter is a

period-dependent-rotation angle measure where the spectral values at each oscillator period are sorted (from

minimum to maximum) for values of rotation angle (𝜃) from 0 to 180 degrees. While the RotI50

can be an adequate measure of the median response, the RotInn measure showed inconsistencies

when calculating the maximum response. Since RotI100 uses a single orientation (rotation angle)

of the ground motion and the maximum response at each period may occur in different orientations,

this measure will not result in the maximum spectral amplitude for all periods, which can be

considered a limitation. On the other hand, RotD100 has the characteristic of resulting in the true

spectral maximum response for each oscillator period as it can be observed in Figure 2.3, which

illustrates the different definitions for a particular pair of recordings of the 1978 Tabas earthquake.

Based on this research, RotDnn was chosen due to the simplicity of its definition and calculation

to be used in the NGA West2 project, and following the same trend as in NGA West1, the median

11

Figure 2.3: Displacement response spectra at 5% damping ratio given by RotDnn and RotInn definitions at 50th and 100th percentiles for the 1978 Tabas earthquake (Boore, 2010).

These decisions generated debate within the earthquake engineering community as to

which definition should be used in the response spectra, the median or the maximum. Stewart et

al. (2011) referred to the BSSC decision as controversial and argued that choices concerning the

spectra definition should be compatible with expected levels of ground motion, expressing that

further research was needed. They supported the use of the maximum definition for structures

with azimuth-independent properties (same stiffness in any direction), but stated that there was no

scientific basis to declare a particular component of ground motion as controlling for

azimuth-dependent structures (different stiffness depending on direction of loading). Given that most

structures would fall on the latter category and not the former, and that differences in stiffness

would mean different fundamental periods of vibration for different azimuths (or principal axes)

12 under seismic loading as the structure would have to align with this particular component. With

this in mind, Nievas and Sullivan (2017) studied the effects of directionality on the behavior of

structures. They focused on the structure typology by evaluating the response of an ‘azimuth

independent’ building (same stiffness in any direction) versus that of an ‘azimuth dependent’

building (See Figure 2.4) to characterize the sensitivity of the geometrical configurations to the

directionality of the motion. While the effects depend on ground motion properties, they also

depend on how sensitive the structure is to them. The authors highlight that, in order to understand

the trends of their proposed framework, hazard curves derived for the maximum definition of the

response spectra, RotD100, are needed given that the RotD50 definition does not provide specific

information with respect to the response in all the possible orientations.

Figure 2.4: Plan views of typologies considered by Nievas and Sullivan (2017).

Furthermore, Shahi and Baker (2014) referred to RotD100 spectrum as an envelope over

spectra from all orientations at each period, which shows agreement with Stewart et al. (2011) with

respect to the shortcoming of RotD100 spectra definition. The maximum response for very similar

periods may occur at very different orientations, which can be deemed to be a very conservative

definition. For this reason, Shahi and Baker also discuss utilizing a response spectrum conditioned

on certain parameters, such as a particular desired orientation for a given period. By choosing a

13 With part of the engineering community moving to use the RotD100 spectra definition for

engineering design, there was a necessity to reconcile the two definitions used, RotD50 (GMPEs

are based on) and RotD100 (desired to use in design practice by some), and be able maintain a

consistency. In 2014, Shahi and Baker developed directionality models that studied the ratio of the

maximum to median component given by the RotDnn definition (𝑆𝑎𝑅𝑜𝑡𝐷100/𝑆𝑎𝑅𝑜𝑡𝐷50). Their goal

was to use these ratios as multiplicative factors to predict the maximum response given that the

NGA West2 ground motion models were developed using the RotD50 definition. The ratios were

computed for 5% damped response spectra for each ground motion of a subset of the NGA West

2 database, and were reported across a discrete range of periods, from 0.01 to 10 seconds as shown

in Table 2.1.

Table 2.1. Geometric mean values of 𝑆𝑎_{𝑅𝑜𝑡𝐷100}/𝑆𝑎_{𝑅𝑜𝑡𝐷50}. Mean and standard deviations for

14 Shahi and Baker compared their proposed directionality model to others and found that

most agreed with each other concerning magnitude and period trends, except for the NEHRP

proposed factors (BSSC 2009) as shown in Figure 2.5. The NEHRP ratios were modelled as the

ratio of the observed maximum intensity (RotD100) to the predicted average intensity

(GMRotD50) by a ground motion model, and not to the observed median value (RotD50), which

is calculated from acceleration data. The research done by Shahi and Baker note that

observed-to-observed ratios provide more numerically stable ratios than observed-to-observed-to-predicted. Given this, for

low periods the new proposed factor of 1.2, and not the previously proposed factor of 1.1, should

be used for low periods in the conversion between definitions for elastic response spectra. The

proposed scale factors were adopted by the NGA West2 Database (Ancheta et al., 2014) to adjust

the 50th percentile spectral response to the maximum.

Figure 2.5: Comparison of models for geometric mean of the ratio SaRotD100/SaRotD50(Shahi & Baker, 2014).

Moreover, Shahi and Baker (2004) found that the ratio of the two definitions was not

influenced significantly by earthquake parameters. They identified the closest distance to fault

15 distance-dependent model and the non-distance dependent model was small. Similarly, Bradley

and Baker (2014) examined the ratios of various response spectra definitions for the Canterbury,

New Zealand earthquake sequence between 2010 and 2011, and found that the ratios of RotD50

to RotD100 were very similar to those obtained by Shahi and Baker (2013). Figure 2.6 illustrates

how the ratios obtained from the Canterbury data compare to the ratios obtained by Shahi and

Baker (2013) that used worldwide data from the NGA West2 database for shallow crustal ground

motions.

Figure 2.6: Comparison of the ratio of 𝑆𝑎_{𝑅𝑜𝑡𝐷100} to 𝑆𝑎_{𝑅𝑜𝑡𝐷50} observed for the Canterbury ground motion
data with the model of Shahi and Baker based on worldwide data (Bradley & Baker, 2014).

**2.1.1 Other Definitions **

Likewise, it is important to understand how response spectra is defined in other countries,

specifically those located in high seismic regions. For instance, in the New Zealand loading

standard, a measure called 𝑆𝑎𝐿𝑎𝑟𝑔𝑒𝑟, specifically defines the response by choosing the larger

component of the pair of recordings (Beyer & Bommer, 2007; NZS 1170.5, 2004). In the case of

16 (2007) assume it is the envelope spectrum since it is the most commonly used in the European

GMPEs. However, directions in the guidelines of EC8 seem to direct the engineer to compute the

response spectra for the three components and consider one or more of the “alternative shapes of

response spectra” for design (EC8-EN1998-1, 2004). Bommer and Ruggeri (2002) reviewed how

acceleration time histories are treated in different seismic codes around the world for buildings

and bridges. They conclude that the guidance provided on how to select the data is inadequate in

all cases. In most cases, how the two horizontal components are treated for the definition of the

response spectrum is not explicitly found in the narrative of the codes (Beyer & Bommer, 2007;

Bommer & Acevedo, 2004). This raises concerns given that there needs to be compatibility

between design methods and hazard representation (Baker & Cornell, 2006). For instance, when

scaling records to a target design spectrum, engineers need to maintain consistency in the definition

of the scaled response spectrum and the target spectrum. Traditionally, definitions such as the

envelope spectrum of the two components and the SRSS (Square Root of the Sum of Squares)

spectrum are used in codes. It is also worth noting, that regardless of the definition of the horizontal

component, the majority of the design spectra in codes are based on uniform hazard spectra (UHS),

where the ordinates have the same frequency of exceedance (Beyer & Bommer, 2007). However,

during verification procedures records should not be scaled to the UHS since the seismic input

would not represent a particular earthquake scenario.

As it is, there is a prevalent lack of awareness as to how response spectra are defined,

especially within the structural engineering community. Therefore, there can be potential impacts

in design if the definition of the hazard is not understood. Fallacies are prone to occur due to the

improper use of measures in seismic design. As the earthquake engineering community moves

17 emphasized. With this in mind, understanding the tools used in seismic design, such as

bidirectional response spectrum, is of upmost importance.

**2.2 Earthquake Data Considerations **

Usually, engineers use simplified methods for the design and assessment of structures, such

as the response spectra method or code-based guidelines. However, some complex situations

require fully dynamic analyses for verification, for example, buildings designed for high ductility

levels, structures with highly irregular configurations or critical structures that, if disrupted, could

have grave impacts. In these situations, the earthquake loading needs to be represented by

acceleration time histories. As such, the criteria for selection of the suite of strong-motion to be

used in these analyses can be critical. Typically, acceleration records that are considered of

engineering interest, meaning those which could cause inelastic structural response, are those

coming from moderate events (5 ≤ 𝑀_{𝑤} < 7), major events (7 ≤ 𝑀_{𝑤} < 8), or great earthquakes

(𝑀𝑤 ≥ 8).